Orthogonal designs V: Orders divisible by eight

Constructions are given for orthogonal designs in orders divisible by eight. These are then used to show all two variable orthogonal designs exist in orders 24, 32 and 48. The existence of two variable designs in order 40 and three variable designs in order 24 is discussed. The conjectures on the existence of all orthogonal designs (1, k) and skew-symmetric weighing matrices for weights k = 1, 2, ..., 2.t9-1 are resolved in the affirmative for orders 2.t9, t > 3 a positive integer.